Utilization of a High-Pressure Vibrating Tube Densimeter for Liquids at Temperatures Down to 100 K

A high-pressure vibrating tube densimeter, specified by the manufacturer for temperatures from (263 to 473) K at pressures up to 140 MPa, was tested at temperatures down to 100 K and from vacuum to pressures up to 10 MPa. To verify the functionality and overall performance under these conditions, the densimeter was calibrated with measurements under vacuum as well as methane and propane as reference fluids. The calibration range is T = (120 to 200) K at pressures from (2.0 to 10.0) MPa. To evaluate the recorded data, two established calibration models were used to describe the dependence of the densimeter's oscillation period on the investigated reference fluids' temperature, pressure, and density. The experiments showed that the vibrating tube densimeter is operational even at temperatures down to 100 K, but exhibits a shift of its vacuum resonance when subjected to thermal cycling at temperatures below 180 K. Accordingly, the calibration models were modified with respect to how the vacuum resonance is considered. Then, the determined calibration parameters reproduce the densities of the reference fluids within ± 0.10 kg·m−3 for the calibration model that performed better for the present study. Measurements on pure ethane and argon validate the calibration of the densimeter. Here, the densities are within (− 0.47 to 0.16) kg·m−3 of values calculated with the respective reference equation of state. The estimated combined expanded uncertainty (k = 2) in density for the validation measurements ranges from (0.52 to 1.13) kg·m−3 or is less than 0.1 % for liquid densities. Supplementary Information The online version of this article (10.1007/s10765-024-03357-9) contains supplementary material, which is available to authorized users.


Experimental Results of the Calibration Measurements (to Section 4.2)
The experimental results reported in this section refer to the modified calibration models where experimental vacuum oscillation periods  0,exp are used instead of the quadratic polynomials that are proposed by the authors of the calibration models [1][2][3].The calibration parameters for the determination of (, ) and (, ) according to both models are given in Table 2 of the paper.

Table S1
Results of the calibration measurements of methane for experimentally determined vacuum oscillation periods  0,exp , where  is the temperature,  is the pressure, and  is the oscillation period. EOS is the density calculated with the reference equations of state by Setzmann and Wagner [4],  May is the density calculated with the modified May et al. [1,2] model, and  OM is the density calculated with the modified Outcalt and McLinden [3]   were corrected with  0 ′ =  0,exp + ( −  0,exp ) •  0  ⁄ .The sensitivity  0  ⁄ is estimated with the temperature-derivative of the respective quadratic fit  0,fit () as given in Table 2 of the paper Table S2 Results of the calibration measurements of propane for experimentally determined vacuum oscillation periods  0,exp , where  is the temperature,  is the pressure, and  is the oscillation period. EOS is the density calculated with the reference equations of state by Lemmon et al. [5],  May is the density calculated with the modified May et al. [1,2]  a Due to temperature deviations between the fluid and vacuum measurements, the values of  0,exp were corrected with  0 ′ =  0,exp + ( −  0,exp ) •  0  ⁄ .The sensitivity  0  ⁄ is estimated with the temperature-derivative of the respective quadratic fit  0,fit () as given in Table 2 of the paper

Results for 𝝉 𝟎 (𝑻) Calculated with Quadratic Polynomials (to Section 4.2)
The experimental results reported in this section refer to the densities determined by using vacuum oscillation periods calculated with quadratic polynomials  0,fit ().The calibration parameters for the determination of (, ) and (, ) according to both models [1][2][3] are given in Table 2 of the paper.
Table S3 Results of the calibration measurements of methane when using the quadratic polynomials  0,fit (), where  is the temperature,  is the pressure, and  is the oscillation period. EOS is the density calculated with the reference equations of state by Setzmann and Wagner [4],  May is the density calculated with the model by May et al. [1,2], and  OM is the density calculated with the model by Outcalt and McLinden [3].∆  are the corresponding deviations from

Table S6
Results of the validation measurements of argon when using the quadratic polynomials  0,fit (), where  is the temperature,  is the pressure, and  is the oscillation period. EOS is the density calculated with the reference equations of state by Tegeler et al. [7],  May is the density calculated with the model by May et al. [1,2], and  OM is the density calculated with the model by Outcalt and McLinden [3] S7 and Table S8.
Due to temperature deviations between the fluid and vacuum measurements, the values of  0,exp

Estimated
Uncertainties in Density (to Section 4.3) In Section 4.3 of the manuscript, an uncertainty analysis has been presented.The uncertainty budget is determined in line with the "Guide to the Expression of Uncertainty in Measurement" [8] (ISO/IEC Guide), referred to as GUM.The combined expanded uncertainty in density () is determined by Equation (10) in the manuscript, covering contributions of the measurements uncertainties of pressure (), temperature () and oscillation period (), as well as the uncertainties resulting from the reproducibility ( repro ) , calibration ( cal ) , reference equations of state ( EOS ) , and the temperature correction of the vacuum oscillation period ( 0,corr ).Here, the contribution of the calibration ( cal ) varies with the used calibration model.It has been conservatively defined as the root of the sum of squared residuals between the experimental densities and the densities calculated with the reference equations of state.Using measured values for the vacuum oscillation period, this contribution is 1.095 kg•m −3 for the modified May et al model and 0.306 kg•m −3 for the modified Outcalt and McLinden model.The resulting individual uncertainties for all (, , ) state points of the validation measurements are listed in Table model.∆  are the corresponding deviations from  EOS model, and  OM is the density calculated with the modified Outcalt and McLinden [3] model.∆  are the corresponding deviations from  EOS

Table S4
[1,2]ts of the calibration measurements of propane when using the quadratic polynomials  0,fit (), where  is the temperature,  is the pressure, and  is the oscillation period.EOS is the density calculated with the reference equations of state by Lemmon et al.[5],  May is the density calculated with the model by May et al.[1,2], and  OM is the density calculated with the model byOutcalt and McLinden [3].∆  are the corresponding deviations from  EOS

Table S5
[1,2]ts of the validation measurements of ethane when using the quadratic polynomials  0,fit (), where  is the temperature,  is the pressure, and  is the oscillation period.EOS is the density calculated with the reference equations of state by Bücker and Wagner[6],  May is the density calculated with the model by May et al.[1,2], and  OM is the density calculated with the model byOutcalt and McLinden [3].∆  are the corresponding deviations from  EOS . ∆  are the corresponding deviations from  EOS

Table S7
[3]2]imental uncertainties of the validation measurements of ethane for experimentally determined vacuum oscillation periods  0,exp , where  is the temperature,  is the pressure, and  EOS is the density calculated with the reference equations of state by Bücker and Wagner[6]for ethane,  May is the experimental density calculated with the modified May et al.[1,2]model, and  OM is the experimental density calculated with the modified Outcalt and McLinden[3]model.( May ) and ( OM ) are the estimated expanded combined uncertainties (k = 2) for the corresponding models

Table S8
[3]2]imental uncertainties of the validation measurements of argon for experimentally determined vacuum oscillation periods  0,exp , where  is the temperature,  is the pressure, and  EOS is the density calculated with the reference equations of state by Tegeler et al.[7],  May is the experimental density calculated with the modified May et al.[1,2]model and  OM is the experimental density calculated with the modified Outcalt and McLinden[3]model.( May ) and ( OM ) are the estimated expanded combined uncertainties (k = 2) for the corresponding models Due to temperature deviations between the fluid and vacuum measurements, the values of  0.exp were corrected with  0 ′ =  0,exp + ( −  0,exp ) •  0  ⁄ .The sensitivity  0  ⁄ is determined approximately with the derivative of the respective quadratic fit  0,fit () a